1 | \section{Sample of collision participants |
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2 | in nuclear collisions.} |
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3 | |
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4 | \subsection{MC procedure to define collision participants.} |
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5 | \hspace{1.0em} The inelastic |
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6 | hadron--nucleus interactions at ultra--relativistic energies are considered |
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7 | as |
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8 | independent hadron--nucleon collisions. It was shown long |
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9 | time ago \cite{CK78} for the hadron--nucleus collision that such a |
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10 | picture can be obtained starting from the Regge--Gribov |
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11 | approach \cite{BT76}, when one assumes that the hadron-nucleus elastic |
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12 | scattering amplitude is a result of reggeon exchanges between the |
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13 | initial hadron and nucleons from target--nucleus. This result leads to |
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14 | simple and efficient MC procedure \cite{Am86} to define |
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15 | the interaction cross sections and the number of the nucleons |
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16 | participating in the inelastic hadron--nucleus collision: |
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17 | \begin{itemize} |
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18 | \item We should randomly distribute |
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19 | $B$ |
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20 | nucleons from the target-nucleus on the impact parameter plane according |
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21 | to the weight function |
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22 | $T([\vec{b}^{B}_{j}])$. This function represents |
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23 | probability density to find sets of the nucleon impact parameters |
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24 | $[\vec{b}^{B}_{j}]$, where |
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25 | $j=1,2,...,B$. |
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26 | \item For each pair of projectile hadron $i$ and target nucleon |
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27 | $j$ with choosen impact parameters $\vec{b}_{i}$ and |
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28 | $\vec{b}^{B}_{j}$ we should check whether they interact inelastically or |
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29 | not using the probability $p_{ij}(\vec{b}_{i}-\vec{b}^{B}_{j},s)$, |
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30 | where $s_{ij}=(p_{i}+p_{j})^2$ is the squared total c.m. energy of the |
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31 | given pair with the $4$--momenta $p_{i}$ and $p_{j}$, respectively. |
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32 | \end{itemize} |
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33 | |
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34 | In the Regge--Gribov approach\cite{BT76} the probability for an inelastic |
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35 | collision of pair of $i$ and $j$ as a function at the squared impact |
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36 | parameter difference $b_{ij}^2=(\vec{ b}_i-\vec{ b}_j^B)^2 $ and $s$ |
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37 | is given by |
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38 | \begin{equation} |
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39 | \label{SP3} |
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40 | p_{ij}(\vec{ b}_i-\vec{ b}_j^B,s)= |
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41 | c^{-1}[1-\exp{\{-2u(b_{ij}^2,s)\}}] = |
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42 | \sum_{n=1}^{\infty}p^{(n)}_{ij}(\vec{ b}_i-\vec{ b}_j^B,s), |
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43 | \end{equation} |
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44 | where |
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45 | \begin{equation} |
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46 | \label{SP4} |
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47 | p^{(n)}_{ij}(\vec{ b}_i-\vec{ b}_j^B,s) |
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48 | =c^{-1}\exp{\{-2u(b_{ij}^2,s)\}} |
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49 | \frac{[2u(b_{ij}^2,s)]^{n}}{n!}. |
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50 | \end{equation} |
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51 | is the probability to find the $n$ cut Pomerons or the probability for |
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52 | $2n$ strings produced in an inelastic hadron-nucleon collision. These |
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53 | probabilities are defined in terms of the (eikonal) amplitude of |
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54 | hadron--nucleon elastic scattering with Pomeron exchange: |
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55 | \begin{equation} |
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56 | \label{SP5}u(b_{ij}^2,s)=\frac{z(s)}{2}\exp (-b_{ij}^2/4\lambda (s)). |
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57 | \end{equation} |
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58 | The quantities $z(s)$ and $\lambda (s)$ are expressed through the |
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59 | parameters of the Pomeron trajectory, $\alpha _P^{^{\prime }}=0.25$ |
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60 | $GeV^{-2}$ and $\alpha _P(0)=1.0808$, and the parameters of the |
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61 | Pomeron-hadron vertex $R_P$ and $\gamma _P$: |
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62 | \begin{equation} |
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63 | \label{SP6}z(s)=\frac{2c\gamma _P}{\lambda (s)}(s/s_0)^{\alpha _P(0)-1} |
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64 | \end{equation} |
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65 | \begin{equation} |
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66 | \label{SP7}\lambda (s)=R_P^2+\alpha _P^{^{\prime }}\ln (s/s_0), |
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67 | \end{equation} |
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68 | respectively, where $s_0$ is a dimensional parameter. |
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69 | |
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70 | In Eqs. (\ref{SP3},\ref{SP4}) the so--called shower enhancement |
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71 | coefficient $c$ is introduced to determine the contribution of |
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72 | diffractive dissociation\cite{BT76}. Thus, the probability for |
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73 | diffractive dissociation of a pair |
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74 | of nucleons can be computed as |
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75 | \begin{equation} |
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76 | \label{SP8}p_{ij}^d(\vec b_i-\vec b_j^B,s)=\frac{c-1}{c}[p_{ij}^{tot}(\vec |
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77 | b_i-\vec b_j^B,s)-p_{ij}(\vec b_i-\vec b_j^B,s)], |
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78 | \end{equation} |
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79 | where |
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80 | \begin{equation} |
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81 | \label{SP9}p_{ij}^{tot}(\vec b_i-\vec b_j^B,s)=(2/c)[1-\exp |
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82 | \{-u(b_{ij}^2,s)\}]. |
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83 | \end{equation} |
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84 | |
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85 | The Pomeron parameters are found from a global fit of the total, |
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86 | elastic, differential elastic and diffractive cross sections of the |
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87 | hadron--nucleon interaction at different energies. |
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88 | |
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89 | For the nucleon-nucleon, pion-nucleon and kaon-nucleon collisions |
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90 | the Pomeron vertex |
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91 | parameters and shower enhancement coefficients are found: |
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92 | $R^{2N}_{P}=3.56$ $GeV^{-2}$, $\gamma^{N}_P=3.96$ |
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93 | $GeV^{-2}$, $s^{N}_{0} = 3.0$ $GeV^{2}$, $c^{N}=1.4$ and |
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94 | $R^{2\pi}_{P} = 2.36$ $GeV^{-2}$, $\gamma^{\pi}_P = 2.17$ $GeV^{-2}$, |
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95 | and $R^{2K}_{P} = 1.96$ |
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96 | $GeV^{-2}$, $\gamma^{K} _P = 1.92$ $GeV^{-2}$, $s^{K}_{0} = 2.3$ |
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97 | $GeV^{2}$, $c^{\pi}=1.8$. |
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98 | |
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99 | \subsection{Separation of hadron diffraction excitation.} |
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100 | |
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101 | \hspace{1.0em}For each pair of target hadron $i$ and projectile |
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102 | nucleon $j$ with choosen impact |
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103 | parameters $\vec{b}_{i}$ and $\vec{b}^{B}_{j}$ we should check |
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104 | whether they interact inelastically or not using the probability |
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105 | \begin{equation} |
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106 | \label{SP14} |
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107 | p^{in}_{ij}(\vec{b}_{i}-\vec{b}^{B}_{j},s)= |
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108 | p_{ij}(\vec{b}_{i}-\vec{b}^{B}_{j},s) |
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109 | + p_{ij}^d(\vec b_i^A-\vec b_j^B,s). |
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110 | \end{equation} |
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111 | If interaction will be realized, then |
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112 | we have to consider it to be diffractive or nondiffractive with probabilities |
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113 | \begin{equation} |
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114 | \label{SP15} |
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115 | \frac{p_{ij}^d(\vec b_i-\vec b_j^B,s)}{p^{in}_{ij} |
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116 | (\vec{b}^{A}_{i}-\vec{b}^{B}_{j},s)} |
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117 | \end{equation} |
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118 | and |
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119 | \begin{equation} |
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120 | \label{SP16} |
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121 | \frac{p_{ij}(\vec b_i-\vec b_j^B,s)}{p^{in}_{ij} |
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122 | (\vec{b}^{A}_{i}-\vec{b}^{B}_{j},s)}. |
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123 | \end{equation} |
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